The verb "let" is a causitive verb. Other causitive verbs are: make, have, got, help. The causative verb "let" is followed by a noun/pronoun and the base form of the verb (which is actually an infinitive with the "to" left off.

What did you think "let" was?? "Let" is also one of those verbs which doesn't need "to" with the following verb. Maybe if I used a synonym it'll be clear: "Don't let your mom cut your hair." "Don't allow your mom to cut your hair." "Don't permit your mom to cut your hair."

It is impossible to have two verbs for on subject , but there as you mentioned the "let " is a verb for for the son and the "cut" is for the mom

let = 讓 Same as in Chinese really.

You can have more than one verb; one of them is the 'real' verb and the rest of them are objects. John stopped smoking. The police saw the thief run away. Don't let your mom cut your hair. So you are the subject, let is the verb, mom is object(noun), cut is object(verb) too, and the rest parts.